# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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notepad report using the Architect is OK, too.

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Course/Section

Exploration 7.2. Bifurcations and Limit Cycles

1. Alter the model in the “Saxophone” submodule of Module 10 by adding a parameter c:

Ó 1 3

li = V, V = —su+ CV-----------V

b

(a) What part of the model does this affect?

(b) How do solutions behave for values of c between 0 and 2, taking

s = b = 1?

(c) As c increases, what happens to the pitch and amplitude?

2. Suppose the model for a simple harmonic oscillator (a linear model),

x = Ó, y = —x

is modified by adding a parameter c:

x = cx + y, y = —x + cy

(a) What happens to the equilibrium point as c goes from —1 to 1?

(b) What happens to the eigenvalues of the matrix of coefficients as c changes from —1 to 1?

132

Exploration 7.2

3. Suppose we further modify the system of Problem 2:

x = cx + y — x(x2 + y2), y = —x + cy — y(x2 + y2)

where —1 < c < 1. Analyze the behavior of the equilibrium point at (0, 0) as c increases from — 1 to 1. How does it compare with the behavior you observed in Problem 2?

Use the model-based pendulum animation in ODE Architect and watch the pendulum gyrate.

You can modify the system for a simple, undamped nonlinear pendulum (see Chapter 10) to produce a torquedpendulum:

x = y, y = — sin( x) + a

Here a represents a torque applied about the axis of rotation of the pendulum arm. Investigate the behavior of this torqued pendulum for the values of a between 0 and 2 by building the model and animating the phase space as a increases. Explain what kind of behavior the pendulum exhibits as a increases; explain the behavior of any equilibrium points you see.

Sweep on the parameter a, and then animate. To animate a graph with multiple trajectories corresponding to different values of a, click on the animate icon below the word “Tools” at the top left of the tools screen.

The motion of a thin, flexible steel beam, affixed to a rigid support over two magnets, can be modeled by Duffing’s equation:

x = y, y = ax — x3

where x represents the horizontal displacement of the beam from the rest position and a is a parameter that is related to the strength of the magnets. Investigate the behavior of this model for — 1 < a < 1. In particular:

(a) Find all equilibrium points and classify them as to type (e.g., center, saddle point), verifying your phase plots with eigenvalue calculations (use ODE Architect for the eigenvalue calculations). Some of your answers will depend on a.

(b) Give a physical interpretation of your answers to Question (a).

(c) What happens to the equilibrium points as the magnets change from weak (a < 0) to strong (a > 0)?

(d) What happens if you add a linear damping term to the model? (Say, y = ax — X — vy.)

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date_______

Course/Section

Exploration 7.3. Higher Dimensions

For example, 0 = 0,

02 = 3, 03 = 1 can be taken as an initial point and so can

01 = 1, 02 = 0,

03 = (10/3)0'5.

SpinningBodies.

Use ODE Architect to draw several distinct trajectories on the ellipsoid of inertia, 0.5(20 + 0 + 3«2) = 6, for system (12).

Choose initial data on the ellipsoid so that the trajectories become the “visible skeleton” of the invisible ellipsoid. What do the trajectories look like? What kind of motion does each represent? You should be able to get a picture that resembles the chapter cover figure and Figure 7.6. Project your 3D graphs onto the 0102-, «2«3-, and m1 «3-planes, and describe whatyou see. Now apply the equilibrium/eigenvalue/eigenvector calculations from ODE Architect to equilibrium points on each of the m1-, m2-, and m3- axes. Describe the results and their correlation with what you saw on the coordinate planes. Now go to the Library file “A Conservative System: The Momentum Ellipsoid” in the folder “Physical Models” and explain what you see in terms of the previous questions in this problem.

2. Exploration 7.1 (Problem 4) gives a predator-prey model where two species, spiders and lizards, prey on flies. Construct a system of three differential equations that includes the prey in the model. You’ll need to represent growth rates and interactions, and you may want to limit population sizes. Make some reasonable assumptions about these parameters. What long-term behavior does your model predict?

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